Lagrangian under transformation

[kaksmet]

Homework Statement

How does the gauge transformation

$A\rightarrow A +\nabla \Psi(r,t) \\ \phi \rightarrow \phi - \frac{1}{c}\frac{\partial \Psi}{\partial t}$

change the Lagrangian and the motion of a single particle moving in an electromagnetic field.

Homework Equations

The Lagrangian before the transformation
$L=\frac{1}{2}mv^{2} -q\phi + q A\cdot v$

The Attempt at a Solution

I get to a solution that looks like

$L' =\frac{1}{2}mv^{2} -q\phi + q A\cdot v + q\frac{d\Phi}{dt} + q\left(\frac{1}{c}-1\right)\frac{\partial \Phi}{\partial t}$
the factor of 1/c could be the problem. I cannot see the motion of the particle changing because of the transformation and I know that the lagrangian can be changed by the time derivative of a function without changing the equations of motion. However because of the factor I still have partial time derivatives left in the solution which troubles me. Anyone know what could be wrong?

 

[nrqed]

Homework Statement

How does the gauge transformation

$A\rightarrow A +\nabla \Psi(r,t) \\ \phi \rightarrow \phi - \frac{1}{c}\frac{\partial \Psi}{\partial t}$

change the Lagrangian and the motion of a single particle moving in an electromagnetic field.

Homework Equations

The Lagrangian before the transformation
$L=\frac{1}{2}mv^{2} -q\phi + q A\cdot v$

The Attempt at a Solution

I get to a solution that looks like

$L' =\frac{1}{2}mv^{2} -q\phi + q A\cdot v + q\frac{d\Phi}{dt} + q\left(\frac{1}{c}-1\right)\frac{\partial \Phi}{\partial t}$
the factor of 1/c could be the problem. I cannot see the motion of the particle changing because of the transformation and I know that the lagrangian can be changed by the time derivative of a function without changing the equations of motion. However because of the factor I still have partial time derivatives left in the solution which troubles me. Anyone know what could be wrong?

do you mean Psi instead of phi in your last two terms?

In any case, it's clear that there is something not right since the units don't make sense.
What is surely happening is that somewhere (probably in the lagrangian) natural units are used so that c was set equal to one. I don't have my books to verify but I am sure this is what happened (it's probably cq phi in the lagrangian). Did you take the two equations (gauge transfos and the lagrangian) from the same book? Even then, some books are not careful and include the factors of c's in some places and not in other places. I would simply set c=1 in your gauge transformation (or double check another reference which would give the lagrangian with the factors of c shown explicitly)

 

[kaksmet]

Thanks nrqed, you nailed it.

 

[Ben Niehoff]

Is this from Goldstein's Classical Mechanics? Look up the errata. I remember doing this problem, and the problem statement in the book has an extraneous factor of c somewhere.